I am confused on how to find a matrix B in triangular form for some linear transformation T over a basis [tex] \{v_1,v_2, v_3\} [/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we are given a minimal polynomial [tex]m(x) = (x+1)^2 (x-2)[/tex].

Do I want to find a basis [tex] \{w_1,w_2\} [/tex] for [tex]null(T+1)^2[/tex] such that [tex](T+1) w_1 = 0[/tex] and [tex](T+1) w_2 \in S(w_1)[/tex]? Is this because [tex](x+1)^2[/tex] has degree two? This is the part I'm not sure about.

For [tex]w_3[/tex], should I just let it be a basis for [tex]null(T-2)[/tex]?

I tried this for a specific transformation T and got the correct matrix B. (I checked the work by computing the matrix S that relates the old basis (v's) to the new basis (w's) and used the relation [tex]B = S^{-1} A S[/tex] where A is the matrix of T.)

Thanks for the help!

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# Triangular Form of a matrix

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